Internal problem ID [13525]
Internal file name [OUTPUT/12697_Friday_February_16_2024_12_11_05_AM_14984917/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.1 (a).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }-4 y=x^{3}} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunT ODE, case c = 0 <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 127
dsolve(diff(y(x),x$2)+x^2*diff(y(x),x)-4*y(x)=x^3,y(x), singsol=all)
\[ y \left (x \right ) = -{\mathrm e}^{-\frac {x^{3}}{3}} \left (\int \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right ) x^{3}d x +\left (-c_{1} -\left (\int \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) x^{3}d x \right )\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right )-c_{2} \right ) \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.589 (sec). Leaf size: 194
DSolve[y''[x]+x^2*y'[x]-4*y[x]==x^3,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {x^3}{3}} \text {HeunT}[4,-2,0,0,-1,x] \left (\int _1^x-\frac {e^{\frac {K[2]^3}{3}} \text {HeunT}[4,0,0,0,1,K[2]] K[2]^3}{\text {HeunT}[4,-2,0,0,-1,K[2]] \text {HeunTPrime}[4,0,0,0,1,K[2]]+\text {HeunT}[4,0,0,0,1,K[2]] \left (\text {HeunT}[4,-2,0,0,-1,K[2]] K[2]^2-\text {HeunTPrime}[4,-2,0,0,-1,K[2]]\right )}dK[2]+c_2\right )+\text {HeunT}[4,0,0,0,1,x] \left (\int _1^x\frac {\text {HeunT}[4,-2,0,0,-1,K[1]] K[1]^3}{\text {HeunT}[4,-2,0,0,-1,K[1]] \text {HeunTPrime}[4,0,0,0,1,K[1]]+\text {HeunT}[4,0,0,0,1,K[1]] \left (\text {HeunT}[4,-2,0,0,-1,K[1]] K[1]^2-\text {HeunTPrime}[4,-2,0,0,-1,K[1]]\right )}dK[1]+c_1\right ) \]